Likelihood ratio tests for many groups in high dimensions

In this paper we investigate the asymptotic distribution of likelihood ratio tests in models with several groups, when the number of groups converges with the dimension and sample size to infinity. We derive central limit theorems for the logarithm of various test statistics and compare our results with the approximations obtained from a central limit theorem using a two step procedure: first consider the number of groups fixed and assume that the sample size and dimension converge to infinity, secondly investigating the resulting distribution if the number of groups converges to infinity.